An Interval Graph-based Tabu Search Framework for Multi-dimensional Packing

Packing is a vast and continuously growing field in the area of Operations Research and Combinatorial Optimization. Roughly speaking, given a set of items and a set of containers, packing problems are concerned with the loading of the items into the containers, according to some packing rules and optimizing a performance measure. Under this broad definition, we find a wide set of real-life and academic problems from the Knapsack problem to cargo loading problems with special balancing constraints to satisfy. In this paper, we consider the packing problems where the number of dimensions of the items and container sizes to consider is at least equal to 2, namely the Multi-Dimensional Orthogonal Packing problems. As reported in several papers and surveys ([6], [3] and [4]), one of the main differences between Mono and Multi-Dimensional Orthogonal Packing problems is the feasibility phase, i.e. given the items assigned to a container and their position in the container according to a set of frame axes, the assignment is feasible if and only if each pair of items does not overlap. We develop a heuristic framework for solving multi-dimensional problems based on the results by Fekete and Shephers [2], that introduced a new implicit representation of a packing in a multi-dimensional environment by a graph-theoretical characterization, The framework splits the performance measure evaluation, that is dependant on the problem, from the packing representation and the feasibility testing, that is common to all the packing problems. More precisely, a two-level Tabu Search (TS) framework is used, where the first level TS changes the assignments of items to containers and optimize the objective function of the packing problem under study,